PROBABILITY

In the realm of probability theory, understanding the terminology related to experiments, sample spaces, events, and random variables is crucial. An experiment consists of potential outcomes, while the sample space encompasses all possible outcomes. Events are subsets of the sample space, and random variables map outcomes to numbers. Through probability distributions, we assign probabilities to different values of random variables, ensuring that the probabilities are between 0 and 1 and sum up to 1 across all possible values. Dive into the fundamental concepts of probability theory through this informative content.

• Probability
• Experiments
• Sample Spaces
• Events
• Random Variables

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1. PROBABILITY David Kauchak CS158 Spring 2022

2. Admin Midterm: back on Thursday Assignment grading update Assignment 6

3. Basic probability theory: terminology An experiment has a set of potential outcomes, e.g., throw a die, look at another example The sample space of an experiment is the set of all possible outcomes, e.g., {1, 2, 3, 4, 5, 6} For machine learning the sample spaces can be very large

4. Basic probability theory: terminology An event is a subset of the sample space Dice rolls {2} {3, 6} even = {2, 4, 6} odd = {1, 3, 5} Machine learning A particular feature has particular values An example, i.e. a particular setting of feature values label = Chardonnay

5. Events We re interested in probabilities of events p({2}) p(label=survived) p(label=Chardonnay) p( Pinot occurred)

6. Random variables A random variable is a mapping from the sample space to a number (think events) It represents all the possible values of something we want to measure in an experiment For example, random variable, X, could be the number of heads for a coin space HHH HHT HTH HTT THH THT TTH TTT X 3 2 2 1 2 1 1 0 Really for notational convenience, since the event space can sometimes be irregular

7. Random variables We re interested in the probability of the different values of a random variable The definition of probabilities over all of the possible values of a random variable defines a probability distribution space HHH HHT HTH HTT THH THT TTH TTT X 3 2 2 1 2 1 1 0 X P(X) 3 P(X=3) = 1/8 2 P(X=2) = 3/8 1 P(X=1) = 3/8 0 P(X=0) = 1/8

8. Probability distribution To be explicit A probability distribution assigns probability values to all possible values of a random variable These values must be >= 0 and <= 1 These values must sum to 1 for all possible values of the random variable X P(X) X P(X) 3 P(X=3) = 1/2 3 P(X=3) = -1 2 P(X=2) = 1/2 2 P(X=2) = 2 1 P(X=1) = 1/2 1 P(X=1) = 0 0 P(X=0) = 1/2 0 P(X=0) = 0

9. Unconditional/prior probability Simplest form of probability is P(X) Prior probability: without any additional information, what is the probability What is the probability of heads? What is the probability of surviving the titanic? What is the probability of a wine review containing the word banana ? What is the probability of a passenger on the titanic being under 21 years old?

10. Joint distribution We can also talk about probability distributions over multiple variables P(X,Y) probability of X and Y a distribution over the cross product of possible values MLPass AND EngPass P(MLPass, EngPass) true, true .88 true, false .01 false, true .04 false, false .07

11. Joint distribution Still a probability distribution all values between 0 and 1, inclusive all values sum to 1 All questions/probabilities of the two variables can be calculate from the joint distribution MLPass AND EngPass P(MLPass, EngPass) true, true .88 What is P(ENGPass)? true, false .01 false, true .04 false, false .07

12. Joint distribution Still a probability distribution all values between 0 and 1, inclusive all values sum to 1 All questions/probabilities of the two variables can be calculate from the joint distribution 0.92 MLPass AND EngPass P(MLPass, EngPass) true, true .88 How did you figure that out? true, false .01 false, true .04 false, false .07

13. Joint distribution This is called summing over or marginalizing out a variable P(x)= p(x,y) y Y MLPass P(MLPass) true 0.89 MLPass AND EngPass P(MLPass, EngPass) false 0.11 EngPass P(EngPass) true, true .88 true, false .01 true 0.92 false, true .04 false 0.08 false, false .07

14. Conditional probability As we learn more information, we can update our probability distribution P(X|Y) models this (read probability of X givenY ) What is the probability of a heads given that both sides of the coin are heads? What is the probability the document is about Chardonnay, given that it contains the word Pinot ? What is the probability of the word noir given that the sentence also contains the word pinot ? Notice that it is still a distribution over the values of X

15. Conditional probability p(X |Y) = ? y x In terms of pior and joint distributions, what is the conditional probability distribution?

16. Conditional probability p(X |Y) =P(X,Y) P(Y) Given that y has happened, in what proportion of those events does x also happen y x

17. Conditional probability p(X |Y) =P(X,Y) P(Y) Given that y has happened, what proportion of those events does x also happen y x MLPass AND EngPass P(MLPass, EngPass) true, true .88 What is: p(MLPass=true | EngPass=false)? true, false .01 false, true .04 false, false .07

18. Conditional probability p(X |Y) =P(X,Y) MLPass AND EngPass P(MLPass, EngPass) P(Y) true, true .88 What is: p(MLPass=true | EngPass=false)? true, false .01 false, true .04 false, false .07 P(true, false)=0.01 = 0.125 P(EngPass= false)=0.01+0.07=0.08 Notice this is very different than p(MLPass=true) = 0.89

19. Both are distributions over X Conditional probability Unconditional/prior probability p(X) p(X |Y) MLPass P(MLPass) MLPass P(MLPass| EngPass=false) true 0.89 true 0.125 false 0.11 false 0.875

20. A note about notation When talking about a particular random variable value, you should technically write p(X=x), etc. However, when it s clear , we ll often shorten it Also, we may also say P(X) or p(x) to generically mean any particular value, i.e. P(X=x) P(true, false)=0.01 = 0.125 P(EngPass= false)=0.01+0.07=0.08

21. Properties of probabilities P(AorB) = ?

22. Properties of probabilities P(AorB) = P(A) + P(B) - P(A,B)

23. Properties of probabilities P( E) = 1 P(E) More generally: Given events E = e1, e2, , en p(ei)=1- p(ej) j=1:n, j i P(E1, E2) P(E1)

24. Chain rule (aka product rule) p(X |Y) =P(X,Y) p(X,Y) =P(X |Y)P(Y) P(Y) We can view calculating the probability of X AND Y occurring as two steps: 1. Y occurs with some probability P(Y) 2. Then, X occurs, given that Y has occurred or you can just trust the math

25. Chain rule p(X,Y,Z) =P(X |Y,Z)P(Y,Z) p(X,Y,Z) =P(X,Y |Z)P(Z) p(X,Y,Z) =P(X |Y,Z)P(Y |Z)P(Z) p(X,Y,Z) =P(Y,Z |X)P(X) p(X1,X2,...,Xn) = ?

26. Applications of the chain rule We saw that we could calculate the individual prior probabilities using the joint distribution p(x) = p(x,y) y Y What if we don t have the joint distribution, but do have conditional probability information: P(Y) P(X|Y) p(x)= p(y)p(x | y) y Y

27. Bayes rule (theorem) p(X |Y) =P(X,Y) p(X,Y) =P(X |Y)P(Y) P(Y) p(Y | X) =P(X,Y) p(X,Y) =P(Y |X)P(X) P(X) p(X |Y) =P(Y | X)P(X) P(Y)

28. Bayes rule Allows us to talk about P(Y|X) rather than P(X|Y) Sometimes this can be more intuitive Why? p(X |Y) =P(Y | X)P(X) P(Y)

29. Bayes rule p(disease | symptoms) For everyone who had those symptoms, how likely is the disease? p(symptoms|disease) For everyone that had the disease, how likely is the symptom? p( label| features ) For all examples that had those features, how likely is that label? p(features | label) For all the examples with that label, how likely is this feature p(cause | effect) vs. p(effect | cause)

30. Gaps V I just won t put these away. direct object These, I just won t put away. filler I just won t put away. gap

31. Gaps What did you put away? gap The socks that I put away. gap

32. Gaps Whose socks did you fold and put away? gap gap Whose socks did you fold ? gap Whose socks did you put away? gap

33. Parasitic gaps These I ll put away without folding . gap gap These I ll put away. gap These without folding . gap

34. Parasitic gaps These I ll put away without folding . gap gap 1. Cannot exist by themselves (parasitic) These I ll put my pants away without folding . gap 2. They re optional These I ll put away without folding them. gap

35. Parasitic gaps http://literalminded.wordpress.com/2009/02/10/do ugs-parasitic-gap/

36. Frequency of parasitic gaps Parasitic gaps occur on average in 1/100,000 sentences Problem: Your friend has developed a machine learning approach to identify parasitic gaps. If a sentence has a parasitic gap, it correctly identifies it 95% of the time. If it doesn t, it will incorrectly say it does with probability 0.005. Suppose we run it on a sentence and the algorithm says it is a parasitic gap, what is the probability it actually is?

37. Prob of parasitic gaps Your friend has developed a machine learning approach to identify parasitic gaps. If a sentence has a parasitic gap, it correctly identifies it 95% of the time. If it doesn t, it will incorrectly say it does with probability 0.005. Suppose we run it on a sentence and the algorithm says it is a parasitic gap, what is the probability it actually is? G = gap T = test positive What question do we want to ask?

38. Prob of parasitic gaps Your friend has developed a machine learning approach to identify parasitic gaps. If a sentence has a parasitic gap, it correctly identifies it 95% of the time. If it doesn t, it will incorrectly say it does with probability 0.005. Suppose we run it on a sentence and the algorithm says it is a parasitic gap, what is the probability it actually is? G = gap T = test positive p(g|t)=?

39. Prob of parasitic gaps Your friend has developed a machine learning approach to identify parasitic gaps. If a sentence has a parasitic gap, it correctly identifies it 95% of the time. If it doesn t, it will incorrectly say it does with probability 0.005. Suppose we run it on a sentence and the algorithm says it is a parasitic gap, what is the probability it actually is? G = gap T = test positive p(g|t) =p(t |g)p(g) p(t) p(t |g)p(g) p(g)p(t |g) g G p(t |g)p(g) = = p(g)p(t |g)+ p(g )p(t |g )

40. Prob of parasitic gaps Your friend has developed a machine learning approach to identify parasitic gaps. If a sentence has a parasitic gap, it correctly identifies it 95% of the time. If it doesn t, it will incorrectly say it does with probability 0.005. Suppose we run it on a sentence and the algorithm says it is a parasitic gap, what is the probability it actually is? G = gap T = test positive p(t |g)p(g) p(g|t)= p(g)p(t |g)+ p(g )p(t |g ) 0.95*0.00001 = 0.00001*0.95+0.99999*0.005 0.002

41. Probabilistic Modeling Model the data with a probabilistic model training data specifically, learn p(features, label) probabilistic model p(features, label) tells us how likely these features and this example are

42. An example: classifying fruit Training data label examples red, round, leaf, 3oz, apple probabilistic model: green, round, no leaf, 4oz, apple p(features, label) yellow, curved, no leaf, 4oz, banana banana green, curved, no leaf, 5oz,

43. Probabilistic models Probabilistic models define a probability distribution over features and labels: probabilistic model: 0.004 yellow, curved, no leaf, 6oz, banana p(features, label)

44. Probabilistic model vs. classifier Probabilistic model: probabilistic model: 0.004 yellow, curved, no leaf, 6oz, banana p(features, label) Classifier: probabilistic model: yellow, curved, no leaf, 6oz banana p(features, label)

45. Probabilistic models: classification Probabilistic models define a probability distribution over features and labels: probabilistic model: 0.004 yellow, curved, no leaf, 6oz, banana p(features, label) Given an unlabeled example: yellow, curved, no leaf, 6oz predict the label How do we use a probabilistic model for classification/prediction?

46. Probabilistic models Probabilistic models define a probability distribution over features and labels: probabilistic model: 0.004 yellow, curved, no leaf, 6oz, banana 0.00002 yellow, curved, no leaf, 6oz, apple p(features, label) For each label, ask for the probability under the model Pick the label with the highest probability

47. Probabilistic model vs. classifier Probabilistic model: probabilistic model: 0.004 yellow, curved, no leaf, 6oz, banana p(features, label) Classifier: probabilistic model: yellow, curved, no leaf, 6oz banana p(features, label) Why probabilistic models?

48. Probabilistic models Probabilities are nice to work with range between 0 and 1 can combine them in a well understood way lots of mathematical background/theory an aside: to get the benefit of probabilistic output you can sometimes calibrate the confidence output of a non- probabilistic classifier Provide a strong, well-founded groundwork Allow us to make clear decisions about things like regularization Tend to be much less heuristic than the models we ve seen Different models have very clear meanings

49. Probabilistic models: big questions Which model do we use, i.e. how do we calculate p(feature, label)? How do train the model, i.e. how do we we estimate the probabilities for the model? How do we deal with overfitting?

50. Same problems weve been dealing with so far Probabilistic models ML in general Which model do we use, i.e. how do we calculate p(feature, label)? Which model do we use (decision tree, linear model, non-parametric) How do train the model, i.e. how to we we estimate the probabilities for the model? How do train the model? How do we deal with overfitting? How do we deal with overfitting?

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